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RIVER RESEARCH AND APPLICATIONS
River. Res. Applic. (2007)
Published online in Wiley InterScience
IDENTIFICATION OF A MINIMAL ADEQUATE MODEL TO DESCRIBETHE BIOMASS DYNAMICS OF RIVER EPILITHON
STEPHANIE BOULETREAU,a* OIHANA IZAGIRRE,b FREDERIC GARABETIAN,a
SABINE SAUVAGE,a ARTURO ELOSEGIb and JOSE-MIGUEL SANCHEZ-PEREZa
a Laboratoire d’ecologie fonctionnelle (EcoLab UMR 5245 CNRS-UPS-INPT), Universite Paul Sabatier,
118 route de Narbonne, 31062 Toulouse cedex 09, Franceb Department of Plant Biology and Ecology, Faculty of Science and Technology, University of the Basque Country,
PO Box 644, 48080 Bilbao, Spain
(www.interscience.wiley.com) DOI: 10.1002/rra.1046
ABSTRACT
The present study sought to identify a minimal adequate model to describe the biomass dynamics of river epilithon, a functionalindicator of river health. Identification of minimal adequate models is particularly necessary in river management, given thereduced number of variables authorities are willing to measure routinely. A model previously developed for epilithon dynamicsin a pre-alpine river was applied to epilithon biomasses recorded in contrasting hydrological, trophic and light conditions atvarious sites in the Aguera stream (Spain) over 3 years (11 case studies). A model selection tool, the Akaike InformationCriterion (AIC), was used to determine the optimal combination of parameters. In nine of 11 case studies, the best modeldescribed epilithon biomass dynamics as the equilibrium between phototrophic growth and discharge-dependent loss andignored light, temperature and nutrient influences. The best adequate minimal model i.e. the model that is the best trade-offbetween goodness-of-fit and model simplicity performed best, in years in which clearly contrasting short low and high waterperiods occurred. During years with less marked hydrodynamics, many other abiotic or biotic processes influenced epilithonbiomass dynamics. In these cases, weaker goodness-of-fit had to be accepted to avoid excessively increasing model complexity.Copyright # 2007 John Wiley & Sons, Ltd.
key words: model selection; model complexity; Akaike Information Criterion; periphyton; epilithic biofilms; stream
Received 10 January 2007; Revised 1 June 2007; Accepted 15 June 2007
INTRODUCTION
Since Streeter and Phelps (1925) started modelling river water quality, ecological models have been increasingly
used by environmental managers, especially from the beginning of the 1970s (Brown and Barnwell, 1987; Even
et al., 1998; Reichert, 2001; Reichert et al., 2001). One fundamental goal of ecological modelling is to predict how
the structure and function of communities respond to change, not only because streams and rivers are naturally
variable, but also because they are vulnerable to anthropogenic disturbances (Power et al., 1988). Awareness of the
importance of a healthy environment has grown steadily during recent decades among environmental management
authorities, leading to new, more ecologically sound policies such as the European Water Framework Directive
(2000/60/EC). Following this new legislation, river health monitoring has been substantially intensified (gauging
stations, water quality sampling), thus providing an important source of hydrological, physical and chemical data
for modelling. Choice of an appropriate set of variables in ecological models could therefore reduce the effort and
cost of data collection for improving fundamental knowledge and decision-making. In this context, it is important
to keep the models as simple as possible without losing predictive power, as complexity hinders the use of models
while not always improving their output.
River health monitoring has concentrated on the use of structural measurements such as the concentrations of
relevant chemicals (nutrients and/or pesticides), invertebrate community composition and algal biomass. Epilithon
has proven to be a reliable indicator of eutrophication (Paul et al., 1991; Rolland et al., 1997; Dodds et al., 1998),
*Correspondence to: Stephanie Bouletreau, Laboratoire d’ecologie fonctionnelle (EcoLab UMR 5245 CNRS-UPS-INPT), Universite PaulSabatier, 118 route de Narbonne, 31062 Toulouse cedex 09, France. E-mail: stephanie.bouletreau@cict.fr
Copyright # 2007 John Wiley & Sons, Ltd.
S. BOULETREAU ET AL.
and the dynamics of epilithic biomass can also be considered a functional indicator of river health (sensu Matthews
et al., 1982) integrating local prevailing conditions with algal development.
A number of models have been designed to simulate the development of river epilithon. Some simple early
models (e.g. McIntire, 1973; Horner et al., 1983; Momo, 1995; Uehlinger et al., 1996; Saravia et al., 1998) related
peak epilithic biomass to environmental variables such as nutrient and light availability, whereas other more
complex models (e.g. Asaeda and Hong Son, 2000; Asaeda and Hong Son, 2001; Flipo et al., 2004) focused on
different component species of epilithon. Uehlinger et al. (1996) presented a model modified from McIntire (1973),
in which they explained the temporal variations in epilithic biomass in a flood-prone pre-alpine river in terms of
photosynthetic accrual and discharge-dependent loss. Its development was based on an experimental dataset for
which sampling strategy (high frequency and long duration) provided a guarantee of better model strength. The
model was later adapted by Bouletreau et al. (2006) to the large Garonne River by adding a term related to
autogenic sloughing. In any case, the level of complexity of epilithon models still used is highly variable and an
issue worthy of more research.
In this paper we address the optimal level of complexity for models of epilithic biomass, taking into account the
trade-off between model complexity and goodness-of-fit, by means of the Akaike Information Criterion (AIC)
(Akaike, 1969; Rawlings et al., 1998), a model selection tool. Our starting hypothesis was that a simple model
could satisfactorily describe epilithon biomass dynamics in a large range of environmental conditions whilst
providing reliable parameters.
METHODS
We assessed the biological realism of a hierarchical family of sub-models based on Uehlinger et al. (1996, Equation
1) according to the possibility of testing the assumptions (that form part of each sub-model) and to the biological
interpretability of the parameters. A twofold approach was adopted in which: (i) competing sub-models with
different combinations of predictor variables were compared and ranked to determine the best sub-model
formulation and (ii) when a minimal adequate model was found to describe epilithon dynamics in the Aguera
stream, the parameters estimated were interpreted.
Model presentation
The hierarchical family of sub-models presented and tested here are derived from the differential equation below
(Equation 1). The complete equation of the model (all processes tested) was
dB
dt¼ mmax;0B
1
1þkinv;BBebðT�T0Þ I
I þ kI
½P�½P�þkP
� cdetQðB � B0Þ � kfloodQðB � B0Þ (1)
with
kflood ¼ kcat if Q > Qcr
kflood ¼ 0 if Q � Qcr
where t is the time (day), B represents the epilithon biomass, T is the water temperature (8C), T0 is the reference
temperature (8C), I is the daily-integrated light intensity (E m�2), Q is the mean daily discharge (m3 s�1) and Qcr is
the critical discharge for the onset of bed load transport (m3 s�1). Uehlinger et al. (1996) developed this equation for
a Swiss pre-alpine gravel bed river (river Necker) characterized by frequent unpredictable disturbances. In that
river, the simplest model acceptably fitting the data employed a biomass-dependent growth rate (mmax,0 and kinv,B), a
detachment rate directly proportional to discharge and biomass (cdet) and a catastrophic loss rate during bed moving
spates (kcat). As the kcat value of 100 day�1 set by Uehlinger et al. (1996) is too high for the Aguera, we evaluated
kcat influence on epilithon dynamics by calibration. Nutrient limitation was implemented in accordance with
Izagirre and Elosegi (2005) and described by a Monod-type rate reduction factor. Only the reduction of phosphorus
[P], the most limiting nutrient, was considered as in Bouletreau et al. (2006).
The state variable B was expressed in grams of ash-free dry matter per surface unit (g AFDM m�2). We opted to
assess biomass dynamics using AFDM, which describes the entire biomass of the assemblage, rather than the
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
A MINIMAL ADEQUATE MODEL FOR RIVER EPILITHON BIOMASS
chlorophyll a, which only relates to the algal part of the mat and can be subject to change because of
photoadaptation. The reference temperature T0 was set to 208C. The initial biomass value corresponded to the
biomass measured at every site at the start of every experimental year. B0 was set to 0, a parameter considered
unnecessary in this case after check. As a result, these parameters were omitted from the calibration.
The differential equation was solved numerically by coding the fourth-order Runge–Kutta in FORTRAN90.
Additional graphics subroutines permitted the main programme to simultaneously display the results. Preliminary
tests had demonstrated that a time step fixed at 3 h is a good prerequisite condition to reduce errors caused by
numerical integration. Values for discharge, temperature, light and phosphate at each time step were obtained by
linear interpolation of observation data.
The simplest sub-model was the linear model with one parameter (mmax,0) and improvements (parameter addition)
were performed step by step. Simulations from every sub-model employing 1–7 parameters were then compared.
Model selection
To adjust the model, we calibrated parameter values that best fitted observed biomass dynamics at each site. The
iterative Marquardt–Levenberg algorithm (Press et al., 1988) was used to minimize the residual sum of squares
(RSS) between modelled and observed biomass for each sub-model at each site. The Akaike information criterion
(AIC) was employed to compare sub-models by quantifying the trade-off between goodness-of-fit (RSS) and model
complexity (number of parameters). The second-order derivative AICc (Equation 2), which contains a bias
correction term for small sample size, was used because the number of free parameters p exceeded approximately n/
40 (where n is the sample size):
AICc ¼ �2lnðLðup=yÞ þ 2p þ 2pð p þ 1Þ
ðn � p � 1Þ (2)
where Lðup=yÞ represents the likelihood of the model parameters given the data y. We computed the AICc using the
following equation (Equation 3):
AICc ¼ nlnRSS
nþ 2p þ 2p
ð p � 1Þðn � p � 1Þ (3)
The AIC penalizes the addition of parameters according to the principles of simplicity and parsimony and thus
selects a model that fits well but has a minimal number of parameters. Second derived measures Di (Equation 4) and
Akaike weight vi (Equation 5) as calculated below were also used to calculate the probability, given the data, of
each sub-model being the best of all those considered (all R models):
Di ¼ AICci � AICcmin (4)
where AICci is the AICc value for model i and AICmin is the AICc value of the best sub-model; and
vi ¼expð�Di=2Þ
PR
r¼1
expð�Dr=2Þ(5)
Evidence ratios were calculated to determine the extent to which the best model ( j) was better than another (i) by
applying vj
�v
i. As a rule of thumb, a Di< 2 (or an evidence ratio < 2.7) suggests substantial evidence for the
model, values between 3 and 7 indicate that the model has considerably less support, whereas a Di> 10 indicates
that the model is very unlikely (Burnham and Anderson, 2002).
Our model approach assumes that the observations are perfect and simulated mechanisms purely deterministic.
These assumptions can lead to bias in parameter estimation and hypothesis testing, as some data series are likely to
include an important stochastic component. Therefore, we only considered data with strong dynamics (sustained or
damped oscillations) and allowing reasonable fits, and deleted from our analyses 1992–93 data from site 7 and
1990–91 data from site 9, which showed a particular pattern likely to be due to development of Sphaerotilus sp.
(unpublished data).
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
S. BOULETREAU ET AL.
Because factors affecting epilithon growth are site- and time-dependent, tests of the adequacy of model structure
were site- and time-specific (Rodriguez, 1987). Simulations were first performed without constraining parameter
values, but in some cases parameter values that minimized the function were biologically unrealistic. Sub-models
were then ranked and compared with simulations with constrained parameters. Realistic constraints on mmax,0, kP, kI
and b were derived for each of the major model parameters, based on field, laboratory and modelling studies
reported in the literature for phytoplanktonic (mainly) and benthic algae. Table I summarizes the parameter values
found in the literature and the constraints we imposed for calibration step. We imposed no constraint on kinv,B and
cdet values, as these parameter values naturally converged towards values calibrated in Uehlinger et al. (1996).
However, the kcat value of 100 days�1 proposed by Uehlinger et al. (1996) was too high to be applied in our work.
Table I. Values of the parameters: mmax,0 (A); kP (B); kI (C) and b (D) obtained from the literature. The constraining range ofparameters used in simulations is given at the end of each sub-section
Description References Values
A. Maximum growth rate (days�1) reported for freshwater/marine phytoplanktonic/benthic algae.Phytoplankton/lake Arhonditsis and Brett (2005 and references therein) 1.2; 1.8; 2.2Phytoplankton/lake Hamilton and Schladow (1997 and references therein) 1.3–3.9Phytoplankton/lake Reynolds (1984 and references therein) 0.21–2.01 (7.97�)Phytoplankton Sterner and Grover (1998) 0.82Phytoplankton/sea Eppley (1972) 2.1Phytoplankton/lake Bouterfas et al. (2002) 1.; 1.64; 1.73Phytoplankton/sea Banse (1982) 0.14–0.7Periphyton Cladophora glomerata/lake Auer and Canale (1982) 0.714; 0.6Periphyton/river Borchardt (1996) 0.8; 2; 2.6–2.7;
0.12–0.47Periphyton/river This study 0.1–2.7
B. Half-saturation constant for phosphorus uptake (mg P L�1) reported for freshwaterphytoplanktonic/benthic algae.
Phytoplankton/lake Arhonditsis and Brett (2005 and references therein) 6; 10; 18Phytoplankton/lake Hamilton and Schladow (1997 and references therein) 1.4–30Phytoplankton/lake Schladow and Hamilton (1997) 1.0–25Phytoplankton/lake Omlin et al. (2001) 1.9Phytoplankton/lake Chen et al. (2002, and reference therein) 0.05; 0.2Phytoplankton/lake Sterner and Grover (1998) 10.1Phytoplankton/marine & freshwater species Lehman et al. (1975) 7.5; 8.7; 9; 12; 15;
16.5; 60–75; 240Periphyton/river Bothwell (1985) 0.5; 0.8; 1.2;
1.6; 2.3; 7.2Periphyton/river Borchardt (1996) 0.3–3; 5–23;
7.5–42; 15–122; 45Periphyton/river This study 0.05–240
C. Light half-saturation constant (mE m�2 s�1) reported for marine phytoplankton.Phytoplankton Klausmeier and Litchman (2001) 50Phytoplankton Huisman et al. (1999) 36Chlorophyte Bates (1976) 5.5–21Coccolithus huxleyi Parsons et al. (1961) 7Ditylum brighwellii Parsons et al. (1961) 29Sargassum sp. Carpenter and Cox (1974) 46Skeletonema costatum Bates (1976) 1.0–21Periphyton/river This study 1.0–50
D. Temperature coefficient (8C�1) reported for freshwater phytoplanktonPhytoplankton/lake Arhonditsis and Brett (2005
and references therein)0.069
Phytoplankton/lake Omlin et al. (2001) 0.046Periphyton/river This study 0.01–0.1
�This measurement was performed in particular temperature conditions (408C).
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
A MINIMAL ADEQUATE MODEL FOR RIVER EPILITHON BIOMASS
Model interpretation
To explain model behaviour and to reduce the freedom as much as possible, we interpreted parameter values of
the best sub-model selected in the first modelling step and used step-wise regression analysis (Statview software) to
explore the influence of environmental factors on these. Only variables with p< 0.05 were kept in the equation. The
explicative factors were mean velocity, percentage of sand, conductivity, total nitrogen, total phosphorus, silicate
and canopy cover.
Experimental data set
We used data on epilithon biomass (g AFDM m�2) in the Aguera stream (Northern Spain) published by Elosegi
and Pozo (1998) and Izagirre and Elosegi (2005). The Aguera is a flood-prone, steep, 30-km long stream draining a
145 km2 basin with humid maritime climate. The number of floods, and the duration of flood-free periods in
particular, can change markedly from year to year (Elosegi et al., 2002; Elosegi et al., 2006). Physico-chemical
characteristics of the water are quite contrasting, reflecting the geology and land-use of different parts of the basin
(Table II). At the headwaters (site 2) conductivity and nutrient contents are low, but they increase sharply when the
stream runs through the villages of Villaverde (site 4) and Trucıos (site 5), decrease again further downstream (site
7) and increase below the town of Guriezo (site 9). Izagirre and Elosegi (2005) showed that at open sites flow is the
main temporal controller, whereas at closed sites the effects of light availability prevail, thus giving more similar
seasonal patterns from year to year.
Epilithon sampling was performed monthly during three 12-month periods (January 1990–January 1991;
October 1992–November 1993; October 2001–November 2002) at five sites (2, 4, 5, 7 and 9). Study cases were
named according to measurement period and site (e.g. case 90–2 was collected in 1990 from site 2). Ten stones were
collected at random in a 100 m2-area in a given riffle at each study site. Study sites are between 2–7 km apart, so we
can therefore reasonably assume that differences in stream bottom facies and biomass levels observed from site to
site ensure the relevance of considering them as distinct study cases. In contrast, in the river Necker studies, the
measured biomasses used for comparison with simulated biomasses were sampled in a reach of 2 km length to
ensure good predictability of the succession sequences of epilithon at a local scale, as epilithon distribution is very
patchy and underdeterministic (Uehlinger et al., 1996).
Additional data used for modelling included daily discharge and solar radiation and periodic data on water
chemistry. Discharge was measured daily by the Spanish Northern Hydrological Confederation at site 9 and
recalculated for the other sites from empirical regressions. Following Elosegi and Pozo (1998), we considered a
discharge of 30 m3 s�1 at site 9 as the critical threshold for flood-induced epilithic sloughing. Solar radiation
(J cm�2) was measured by the Spanish Meteorological Institute at San Sebastian, and in some periods of missing
data it was calculated from duration of sunshine in Bilbao, by means of the Angstrom formula, which relates solar
radiation to extraterrestrial radiation and relative sunshine duration (Food and Agricultural Organization of United
States: http://www.fao.org/). Every (measured or calculated) daily radiation value was first converted to PAR
(J cm�2) according to Steemann-Nielsen (1975) and then converted to photon flux density expressed as E m�2. To
calculate the radiation effectively reaching the stream, we corrected the above data for canopy cover. Canopy cover
was measured using 10 vertical photographs taken at each sampling site with a wide-angle lens in winter and
summer foliage. Photographs were scanned, their contrast increased to produce black (covered) and white
(uncovered sky) images, and analysed with NIH 1.55 software (Izagirre and Elosegi, 2005). Water chemistry was
measured approximately every 15 days during 1990–91 and 2001–02, but not during 1992–93. Phosphate was
measured by the stannous chloride method (APHA, 1992) on a Shimadzu UV-1603 spectrophotometer.
RESULTS
Results of AIC statistics for each sub-model at every study site are listed in Table III. The set of candidate
sub-models (R) varied between years: in 2001 R was 7; in 1990 R was 6 because discharge was always lower than
critical and subsequently kcat was never activated; and in 1992 R was 5, as phosphate and temperature data were not
available and thus kP and b were not activated. One single sub-model was clearly the best of all: in 7 of 11 study
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
Tab
leII
.G
eogra
phic
al,physi
cal
and
chem
ical
char
acte
rist
ics
of
the
study
site
s(m
ean�
SD
).W
hen
dif
fere
nt
figure
sap
pea
rse
par
ated
by
a/
sign,th
efi
rst
corr
esponds
toth
eper
iod
1990–91,
the
seco
nd
to2001–02
Sit
eS
ite
2S
ite
4S
ite
5S
ite
7S
ite
9
Ord
er2
33
33
Wat
erte
mper
ature
(8C
)11.1
(�4.1
)/12.5
(�4.6
)11.3
(�4.7
)/13.4
(�4.3
)12.7
(�4.0
)/13.8
(�4.3
)13.3
(�3.9
)/14.2
(�3.5
)13.3
(�3.7
)/15.0
(�4.3
)pH
7.7
(�0.4
)/7.4
(�0.2
)7.9
(�0.5
)/7.5
(�0.3
)8.4
(�0.7
)/7.8
(�0.3
)8.6
(�0.5
)/7.8
(�0.2
)7.8
(�0.4
)/7.5
(�0.2
)C
onduct
ivit
y(m
Scm
�1)
172(�
48)/
148(�
42)
243(�
86)/
203(�
74)
308(�
69)/
248(�
59)
262(�
31)/
234(�
34)
234(�
50)/
216(�
37)
Tota
lN
(mg
NL�
1)
646(�
314)/
493(�
284)
940(�
374)/
747(�
361)
950(�
207)/
1161(�
323)
722(�
322)/
914(�
282)
783(�
302)/
839(�
306)
Tota
lP
(mg
PL�
1)
6(�
10)/
8.5
(�11)
128(�
134)/
69(�
64)
111(�
62)/
105(�
99)
20(�
12)/
20(�
23)
103(�
85)/
38(�
31)
Sil
icat
e(m
gS
iL�
1)
2504(�
898)/
3590(�
477)
2021(�
858)/
2749(�
615)
1793(�
734)/
2572(�
691)
963(�
773)/
2006(�
623)
1275(�
623)/
2036(�
582)
Nit
rate
(mg
NL�
1)
590(�
302)/
443(�
281)
806(�
312)/
618(�
365)
830(�
231)/
1013(�
312)
696(�
327)/
848(�
282)
708(�
259)/
774(�
310)
Nit
rite
(mg
NL�
1)
2(�
1.5
)/17(�
12)
31(�
26)/
24(�
14)
32(�
26)/
34(�
18)
5(�
2)/
26(�
14)
35(�
25)/
20(�
9)
Am
oniu
m(m
gN
L�
1)
54(�
80)/
33(�
16)
104(�
112)/
105(�
77)
88(�
56)/
114(�
89)
21(�
17)/
39(�
9)
90(�
82)/
46(�
9)
Vel
oci
ty(m
s�1)
0.7
7(�
0.1
9)/
0.7
5(�
0.3
1)
0.2
1(�
0.2
6)/
0.5
9(�
0.3
0)
0.9
4(�
0.2
1)/
0.9
7(�
0.4
0)
0.8
2(�
0.1
9)/
0.8
1(�
0.3
3)
0.8
3(�
0.3
3)/
0.8
1(�
0.3
9)
San
d(%
)5.0
/25
23/0
2.0
/14
4.7
/05.8
/0C
anopy
cover
(%)
87/6
53.0
/72
8.0
/37
31/3
277/7
3
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
S. BOULETREAU ET AL.
Tab
leII
I.A
kai
ke
info
rmat
ion
stat
isti
csfo
rdif
fere
nt
sub-m
odel
s
Act
ivat
edpro
cess
esn
pR
SS
AIC
cD
AIC
cex
p(-D
AIC
c/2)
vi
vj/v
i
Stu
dy
case
:90–2
Max
.gro
wth
Hydro
det
ach
13
2521.2
53.1
91.36
0.5
10.1
52.0
Max.
gro
wth
Hyd
rodet
ach
Thic
knes
s13
3359.6
51.83
0.00
1.0
00.3
01.0
Max
.gro
wth
Hydro
det
ach
T8C
13
3521.2
56.6
54.8
30.0
90.0
311.2
Max
.gro
wth
Hydro
det
ach
Lig
ht
13
3379.0
52.5
10.68
0.7
10.2
11.4
Max
.gro
wth
Hydro
det
ach
Nutr
ient
13
3419.9
53.8
42.0
20.3
60.1
12.7
Max
.gro
wth
Hydro
det
ach
Thic
knes
sT8C
13
4339.3
55.4
13.5
80.1
70.0
56.0
Max
.gro
wth
Hydro
det
ach
Thic
knes
sL
ight
13
4349.3
55.7
83.9
60.1
40.0
47.2
Max
.gro
wth
Hydro
det
ach
Thic
knes
sN
utr
ient
13
4359.6
56.1
64.3
30.1
10.0
38.7
Max
.gro
wth
Hydro
det
ach
T8C
Lig
ht
13
4376.9
56.7
74.9
40.0
80.0
311.8
Max
.gro
wth
Hydro
det
ach
T8C
Nutr
ient
13
4392.0
57.2
85.4
60.0
70.0
215.3
Max
.gro
wth
Hydro
det
ach
Lig
ht
Nutr
ient
13
4376.9
56.7
74.9
40.0
80.0
311.8
Max
.gro
wth
Hydro
det
ach
Thic
knes
sT8C
Lig
ht
13
5339.3
60.9
89.1
50.0
10.0
097.1
Max
.gro
wth
Hydro
det
ach
Thic
knes
sT8C
Nutr
ient
13
5339.3
60.9
89.1
50.0
10.0
097.1
Max
.gro
wth
Hydro
det
ach
Thic
knes
sL
ight
Nutr
ient
13
5349.3
61.3
59.5
30.0
10.0
0117.2
Stu
dy
case
:90–4
Max.
gro
wth
Hyd
rodet
ach
Thic
knes
s13
3495.0
55.98
0.00
1.0
00.7
31.0
Max
.gro
wth
Hydro
det
ach
Thic
knes
sT8C
13
4495.0
60.3
14.3
30.1
10.0
88.7
Max
.gro
wth
Hydro
det
ach
Thic
knes
sL
ight
13
4495.0
60.3
14.3
30.1
10.0
88.7
Max
.gro
wth
Hydro
det
ach
Thic
knes
sN
utr
ient
13
4495.0
60.3
14.3
30.1
10.0
88.7
Max
.gro
wth
Hydro
det
ach
Thic
knes
sT8C
Lig
ht
13
5495.0
65.8
99.9
00.0
10.0
1141.5
Max
.gro
wth
Hydro
det
ach
Thic
knes
sT8C
Nutr
ient
13
5495.0
65.8
99.9
00.0
10.0
1141.5
Max
.gro
wth
Hydro
det
ach
Thic
knes
sL
ight
Nutr
ient
13
5495.0
65.8
99.9
00.0
10.0
1141.5
Stu
dy
case
:90–5
Max.
gro
wth
Hyd
rodet
ach
Thic
knes
s13
312203.7
97.65
0.00
1.0
00.7
31.0
Max
.gro
wth
Hydro
det
ach
Thic
knes
sT8C
13
412203.7
101.9
84.3
30.1
10.0
88.7
Max
.gro
wth
Hydro
det
ach
Thic
knes
sL
ight
13
412203.7
101.9
84.3
30.1
10.0
88.7
Max
.gro
wth
Hydro
det
ach
Thic
knes
sN
utr
ient
13
412203.6
101.9
84.3
30.1
10.0
88.7
Max
.gro
wth
Hydro
det
ach
Thic
knes
sT8C
Lig
ht
13
512203.6
107.5
59.9
00.0
10.0
1141.5
Max
.gro
wth
Hydro
det
ach
Thic
knes
sT8C
Nutr
ient
13
512203.6
107.5
59.9
00.0
10.0
1141.5
Max
.gro
wth
Hydro
det
ach
Thic
knes
sL
ight
Nutr
ient
13
512203.6
107.5
59.9
00.0
10.0
1141.5
Stu
dy
case
:90–7
Max
.gro
wth
13
18928.0
87.2
84.3
30.1
10.0
68.7
Max
.gro
wth
Thic
knes
s13
28928.0
90.1
27.1
60.0
30.0
135.9
Max
.gro
wth
T8C
13
28928.0
90.1
27.1
60.0
30.0
135.9
Max
.gro
wth
Lig
ht
13
28928.0
90.1
27.1
60.0
30.0
135.9
Max
.gro
wth
Nutr
ient
13
28928.0
90.1
27.1
60.0
30.0
135.9
Max
.gro
wth
Hydro
det
ach
13
27111.2
87.1
64.2
00.1
20.0
68.2
Max.
gro
wth
Thic
knes
sH
ydro
det
ach
13
33941.5
82.95
0.00
1.0
00.5
21.0
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007
DOI: 10.1002/rra
A MINIMAL ADEQUATE MODEL FOR RIVER EPILITHON BIOMASS
(Conti
nues
)
)
Tab
leII
I.(C
onti
nued
)
Act
ivat
edpro
cess
esn
pR
SS
AIC
cD
AIC
cex
p(-D
AIC
c/2)
vi
vj/v
i
Max
.gro
wth
T8C
Hydro
det
ach
13
37111.2
90.6
27.6
70.0
20.0
146.3
Max
.gro
wth
Lig
ht
Hydro
det
ach
13
35695.5
87.7
44.7
90.0
90.0
510.9
Max
.gro
wth
Nutr
ient
Hydro
det
ach
13
36176.7
88.7
95.8
40.0
50.0
318.5
Max
.gro
wth
Thic
knes
sT8C
Hydro
det
ach
13
43941.5
87.2
94.3
30.1
10.0
68.7
Max
.gro
wth
Thic
knes
sL
ight
Hydro
det
ach
13
43941.5
87.2
94.3
30.1
10.0
68.7
Max
.gro
wth
Thic
knes
sN
utr
ient
Hydro
det
ach
13
43941.5
87.2
94.3
30.1
10.0
68.7
Max
.gro
wth
T8C
Lig
ht
Hydro
det
ach
13
45961.3
92.6
79.7
10.0
10.0
0128.5
Max
.gro
wth
T8C
Nutr
ient
Hydro
det
ach
13
45537.2
91.7
18.7
50.0
10.0
179.5
Max
.gro
wth
Lig
ht
Nutr
ient
Hydro
det
ach
13
45156.7
90.7
87.8
30.0
20.0
150.1
Max
.gro
wth
Thic
knes
sT8C
Lig
ht
Hydro
det
ach
13
53941.5
92.8
69.9
00.0
10.0
0141.5
Max
.gro
wth
Thic
knes
sT8C
Nutr
ient
Hydro
det
ach
13
53941.5
92.8
69.9
00.0
10.0
0141.5
Max
.gro
wth
Thic
knes
sL
ight
Nutr
ient
Hydro
det
ach
13
53941.5
92.8
69.9
00.0
10.0
0141.5
Stu
dy
case
:92–5
Max
.gro
wth
Thic
knes
s13
22303.1
72.5
05.8
70.0
50.0
318.8
Max
.gro
wth
Lig
ht
13
22444.5
73.2
86.6
50.0
40.0
227.8
Max.
gro
wth
Thic
knes
sH
ydro
det
ach
13
31122.9
66.63
0.00
1.0
00.6
41.0
Max
.gro
wth
Thic
knes
sL
ight
13
32303.1
75.9
79.3
40.0
10.0
1106.6
Max
.gro
wth
Thic
knes
sC
ata
det
ach
13
32303.1
75.9
79.3
40.0
10.0
1106.6
Max
.gro
wth
Thic
knes
sL
ight
Hydro
det
ach
13
41122.9
70.9
64.3
30.1
10.0
78.7
Max
.gro
wth
Thic
knes
sC
ata
det
ach
Hydro
det
ach
13
4973.8
69.1
12.4
80.2
90.1
83.5
Max
.gro
wth
Thic
knes
sL
ight
Cat
adet
ach
13
41304.2
72.9
16.2
80.0
40.0
323.1
Max
.gro
wth
Thic
knes
sL
ight
Hydro
det
ach
13
5973.8
74.6
88.0
50.0
20.0
156.1
Stu
dy
case
:92–9
Max
.gro
wth
13
12649.3
71.4
99.6
30.0
10.0
0123.6
Max
.gro
wth
Thic
knes
s13
21188.2
63.9
02.0
50.3
60.2
02.8
Max
.gro
wth
Lig
ht
13
21515.2
67.0
65.2
10.0
70.0
413.5
Max.
gro
wth
Thic
knes
sH
ydro
det
ach
13
3777.6
61.85
0.00
1.0
00.5
51.0
Max
.gro
wth
Thic
knes
sL
ight
13
31188.2
67.3
65.5
10.0
60.0
315.7
Max
.gro
wth
Thic
knes
sC
ata
det
ach
13
31188.2
67.3
65.5
10.0
60.0
315.7
Max
.gro
wth
Lig
ht
Hydro
det
ach
13
31515.2
70.5
28.6
70.0
10.0
176.4
Max
.gro
wth
Lig
ht
Cat
adet
ach
13
31515.2
70.5
28.6
70.0
10.0
176.4
Max
.gro
wth
Thic
knes
sL
ight
Hydro
det
ach
13
4777.6
66.1
94.3
30.1
10.0
68.7
Max
.gro
wth
Thic
knes
sC
ata
det
ach
Hydro
det
ach
13
4800.3
66.5
64.7
10.0
90.0
510.5
Max
.gro
wth
Thic
knes
sL
ight
Cat
adet
ach
13
41188.2
71.7
09.8
50.0
10.0
0137.4
Max
.gro
wth
Thic
knes
sL
ight
Cat
adet
ach
Hydro
det
ach
13
5777.6
71.7
69.9
00.0
10.0
0141.5
Stu
dy
case
:01–2
Max.
gro
wth
Hyd
rodet
ach
13
2209.1
41.31
0.00
1.0
00.2
51.0
Max
.gro
wth
Cat
adet
ach
13
2323.4
46.9
85.6
70.0
60.0
117.0
Max
.gro
wth
Thic
knes
sH
ydro
det
ach
13
3160.5
41.3
40.03
0.9
90.2
41.0
(Conti
nues
)
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007
DOI: 10.1002/rra
S. BOULETREAU ET AL.
)
Tab
leII
I.(C
onti
nued
)
Act
ivat
edpro
cess
esn
pR
SS
AIC
cD
AIC
cex
p(-D
AIC
c/2)
vi
vj/v
i
Max
.gro
wth
Thic
knes
sC
ata
det
ach
13
3323.4
50.4
59.1
30.0
10.0
096.3
Max
.gro
wth
T8C
Hydro
det
ach
13
3208.5
44.7
43.4
30.1
80.0
45.5
Max
.gro
wth
T8C
Cat
adet
ach
13
3323.4
50.4
59.1
30.0
10.0
096.3
Max
.gro
wth
Lig
ht
Hydro
det
ach
13
3199.4
44.1
62.8
50.2
40.0
64.2
Max
.gro
wth
Lig
ht
Cat
adet
ach
13
3323.4
50.4
59.1
30.0
10.0
096.3
Max
.gro
wth
Nutr
ient
Hydro
det
ach
13
3209.1
44.7
83.4
70.1
80.0
45.7
Max
.gro
wth
Nutr
ient
Cat
adet
ach
13
3323.4
50.4
59.1
30.0
10.0
096.3
Max
.gro
wth
Cat
adet
ach
Hydro
det
ach
13
3209.1
44.7
83.4
70.1
80.0
45.7
Max
.gro
wth
Thic
knes
sT8C
Hydro
det
ach
13
4160.5
45.6
74.3
60.1
10.0
38.9
Max
.gro
wth
Thic
knes
sL
ight
Hydro
det
ach
13
4158.9
45.5
54.2
30.1
20.0
38.3
Max
.gro
wth
Thic
knes
sN
utr
ient
Hydro
det
ach
13
4160.5
45.6
74.3
60.1
10.0
38.9
Max
.gro
wth
Thic
knes
sC
ata
det
ach
Hydro
det
ach
13
4127.6
42.6
91.3
80.5
00.1
22.0
Max
.gro
wth
T8C
Lig
ht
Hydro
det
ach
13
4209.1
49.1
17.8
00.0
20.0
149.4
Max
.gro
wth
T8C
Nutr
ient
Hydro
det
ach
13
4209.1
49.1
17.8
00.0
20.0
149.4
Max
.gro
wth
T8C
Hydro
det
ach
Cat
adet
ach
13
4209.1
49.1
17.8
00.0
20.0
149.4
Max
.gro
wth
Lig
ht
Nutr
ient
Hydro
det
ach
13
4209.1
49.1
17.8
00.0
20.0
149.4
Max
.gro
wth
Lig
ht
Hydro
det
ach
Cat
adet
ach
13
4209.1
49.1
17.8
00.0
20.0
149.4
Max
.gro
wth
Thic
knes
sH
ydro
det
ach
Cat
adet
ach
13
4209.1
49.1
17.8
00.0
20.0
149.4
Max
.gro
wth
Thic
knes
sC
ata
det
ach
T8C
13
5127.6
48.2
66.9
50.0
30.0
132.3
Max
.gro
wth
Thic
knes
sC
ata
det
ach
Lig
ht
13
5127.6
48.2
66.9
50.0
30.0
132.3
Max
.gro
wth
Thic
knes
sC
ata
det
ach
Nutr
ient
13
5127.6
48.2
66.9
50.0
30.0
132.3
Max
.gro
wth
Thic
knes
sT8C
Lig
ht
13
5127.6
48.2
66.9
50.0
30.0
132.3
Max
.gro
wth
Thic
knes
sT8c
Nutr
ient
13
5127.6
48.2
66.9
50.0
30.0
132.3
Max
.gro
wth
Thic
knes
sL
ight
Nutr
ient
13
5127.6
48.2
66.9
50.0
30.0
132.3
Max
.gro
wth
T8C
Lig
ht
Nutr
ient
13
5127.6
48.2
66.9
50.0
30.0
132.3
Stu
dy
case
:01–4
Max.
gro
wth
13
1874.9
57.08
0.00
1.0
00.1
41.0
Max
.gro
wth
Thic
knes
s13
2858.2
59.6
72.5
90.2
70.0
43.6
Max
.gro
wth
T8C
13
2874.9
59.9
22.8
40.2
40.0
34.1
Max
.gro
wth
Lig
ht
13
2874.9
59.9
22.8
40.2
40.0
34.1
Max
.gro
wth
Nutr
ient
13
2857.4
59.6
62.5
70.2
80.0
43.6
Max
.gro
wth
Hydro
det
ach
13
2732.7
57.6
10.5
30.7
70.1
11.3
Max
.gro
wth
Cat
adet
ach
13
2866.7
59.8
02.7
10.2
60.0
43.9
Max
.gro
wth
Thic
knes
sH
ydro
det
ach
13
3732.8
61.0
84.0
00.1
40.0
27.4
Max
.gro
wth
Thic
knes
sC
ata
det
ach
13
3614.4
58.7
91.7
10.4
30.0
62.3
Max
.gro
wth
Thic
knes
sT8C
13
3874.9
63.3
96.3
00.0
40.0
123.4
Max
.gro
wth
Thic
knes
sL
ight
13
3874.9
63.3
96.3
00.0
40.0
123.4
Max
.gro
wth
Thic
knes
sN
utr
ient
13
3857.4
63.1
26.0
40.0
50.0
120.5
Max
.gro
wth
T8C
Hydro
det
ach
13
3732.7
61.0
84.0
00.1
40.0
27.4
(Conti
nues
)
Tab
leII
I.(C
onti
nued
)
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
A MINIMAL ADEQUATE MODEL FOR RIVER EPILITHON BIOMASS
Tab
leII
I.(C
onti
nued
)
Act
ivat
edpro
cess
esn
pR
SS
AIC
cD
AIC
cex
p(-D
AIC
c/2)
vi
vj/v
i
Max
.gro
wth
T8C
Lig
ht
13
3874.9
63.3
96.3
00.0
40.0
123.4
Max
.gro
wth
T8C
Nutr
ient
13
3857.4
63.1
26.0
40.0
50.0
120.5
Max
.gro
wth
T8C
Cat
adet
ach
13
3642.8
59.3
82.2
90.3
20.0
43.1
Max
.gro
wth
Lig
ht
Hydro
det
ach
13
3732.7
61.0
84.0
00.1
40.0
27.4
Max
.gro
wth
Lig
ht
Nutr
ient
13
3857.4
63.1
26.0
40.0
50.0
120.5
Max
.gro
wth
Lig
ht
Cat
adet
ach
13
3668.9
59.9
02.8
10.2
40.0
34.1
Max
.gro
wth
Nutr
ient
Hydro
det
ach
13
3732.7
61.0
84.0
00.1
40.0
27.4
Max
.gro
wth
Nutr
ient
Cat
adet
ach
13
3644.7
59.4
22.3
30.3
10.0
43.2
Max
.gro
wth
Cat
adet
ach
Hydro
det
ach
13
3638.8
59.3
02.2
10.3
30.0
53.0
Max
.gro
wth
Thic
knes
sT8C
Cat
adet
ach
13
4607.8
58.6
51.5
70.4
60.0
62.2
Max
.gro
wth
Thic
knes
sL
ight
Hydro
det
ach
13
4732.7
65.4
18.3
30.0
20.0
064.4
Max
.gro
wth
Thic
knes
sL
ight
Cat
adet
ach
13
4607.8
58.6
51.5
70.4
60.0
62.2
Max
.gro
wth
Thic
knes
sN
utr
ient
Hydro
det
ach
13
4732.7
65.4
18.3
30.0
20.0
064.4
Max
.gro
wth
Thic
knes
sN
utr
ient
Cat
adet
ach
13
4607.8
58.6
51.5
70.4
60.0
62.2
Max
.gro
wth
Thic
knes
sC
ata
det
ach
Hydro
det
ach
13
4614.4
63.1
26.0
40.0
50.0
120.5
Max
.gro
wth
T8C
Lig
ht
Hydro
det
ach
13
4732.7
65.4
18.3
30.0
20.0
064.4
Max
.gro
wth
T8C
Lig
ht
Cat
adet
ach
13
4642.8
63.7
16.6
30.0
40.0
127.5
Max
.gro
wth
T8C
Nutr
ient
Hydro
det
ach
13
4732.7
65.4
18.3
30.0
20.0
064.4
Max
.gro
wth
T8C
Nutr
ient
Cat
adet
ach
13
4642.8
63.7
16.6
30.0
40.0
127.5
Max
.gro
wth
T8C
Hydro
det
ach
Cat
adet
ach
13
4638.8
63.6
36.5
50.0
40.0
126.4
Max
.gro
wth
Lig
ht
Nutr
ient
Hydro
det
ach
13
4732.7
65.4
18.3
30.0
20.0
064.4
Max
.gro
wth
Lig
ht
Nutr
ient
Cat
adet
ach
13
4642.8
63.7
16.6
30.0
40.0
127.5
Max
.gro
wth
Lig
ht
Hydro
det
ach
Cat
adet
ach
13
4638.8
63.6
36.5
50.0
40.0
126.4
Max
.gro
wth
Nutr
ient
Hydro
det
ach
Cat
adet
ach
13
4638.8
63.6
36.5
50.0
40.0
126.4
Stu
dy
case
:01-5
Max.
gro
wth
13
12762.0
72.03
0.00
1.0
00.2
91.0
Max
.gro
wth
Thic
knes
s13
22762.0
74.8
62.8
40.2
40.0
74.1
Max
.gro
wth
T8C
13
22762.0
74.8
62.8
40.2
40.0
74.1
Max
.gro
wth
Lig
ht
13
22762.0
74.8
62.8
40.2
40.0
74.1
Max
.gro
wth
Nutr
ient
13
22762.0
74.8
62.8
40.2
40.0
74.1
Max
.gro
wth
Hydro
det
ach
13
22762.0
74.8
62.8
40.2
40.0
74.1
Max
.gro
wth
Cat
adet
ach
13
22762.0
74.8
62.8
40.2
40.0
74.1
Max
.gro
wth
Thic
knes
sH
ydro
det
ach
13
32279.5
75.8
33.8
10.1
50.0
46.7
Max
.gro
wth
Thic
knes
sC
ata
det
ach
13
32762.0
78.3
36.3
00.0
40.0
123.4
Max
.gro
wth
Thic
knes
sT8C
13
32762.0
78.3
36.3
00.0
40.0
123.4
Max
.gro
wth
Thic
knes
sL
ight
13
32762.0
78.3
36.3
00.0
40.0
123.4
Max
.gro
wth
Thic
knes
sN
utr
ient
13
32762.0
78.3
36.3
00.0
40.0
123.4
Max
.gro
wth
T8C
Hydro
det
ach
13
32762.0
78.3
36.3
00.0
40.0
123.4
Max
.gro
wth
T8C
Lig
ht
13
32762.0
78.3
36.3
00.0
40.0
123.4
(Conti
nues
)
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
S. BOULETREAU ET AL.
Tab
leII
I.(C
onti
nued
)
Act
ivat
edpro
cess
esn
pR
SS
AIC
cD
AIC
cex
p(-D
AIC
c/2)
vi
vj/v
i
Max
.gro
wth
T8C
Nutr
ient
13
32762.0
78.3
36.3
00.0
40.0
123.4
Max
.gro
wth
T8C
Cat
adet
ach
13
32762.0
78.3
36.3
00.0
40.0
123.4
Max
.gro
wth
Lig
ht
Hydro
det
ach
13
32491.7
76.9
94.9
60.0
80.0
212.0
Max
.gro
wth
Lig
ht
Nutr
ient
13
32762.0
78.3
36.3
00.0
40.0
123.4
Max
.gro
wth
Lig
ht
Cat
adet
ach
13
32762.0
78.3
36.3
00.0
40.0
123.4
Max
.gro
wth
Nutr
ient
Hydro
det
ach
13
32762.0
78.3
36.3
00.0
40.0
123.4
Max
.gro
wth
Nutr
ient
Cat
adet
ach
13
32762.0
78.3
36.3
00.0
40.0
123.4
Max
.gro
wth
Cat
adet
ach
Hydro
det
ach
13
32279.1
75.8
33.8
00.1
50.0
46.7
Max
.gro
wth
Thic
knes
sL
ight
Hydro
det
ach
13
42215.3
79.8
07.7
70.0
20.0
148.6
Max
.gro
wth
Thic
knes
sN
utr
ient
Hydro
det
ach
13
42279.1
80.1
78.1
40.0
20.0
158.6
Max
.gro
wth
Thic
knes
sC
ata
det
ach
Hydro
det
ach
13
42279.1
80.1
78.1
40.0
20.0
158.5
Stu
dy
case
:01-7
Max
.gro
wth
Hydro
det
ach
13
22249.6
72.2
07.4
80.0
20.0
242.0
Max
.gro
wth
Cat
adet
ach
13
22255.5
72.2
37.5
10.0
20.0
142.7
Max.
gro
wth
Thic
knes
sH
ydro
det
ach
13
3969.5
64.72
0.00
1.0
00.6
31.0
Max
.gro
wth
Lig
ht
Hydro
det
ach
13
31746.2
72.3
77.6
50.0
20.0
145.8
Max
.gro
wth
Nutr
ient
Hydro
det
ach
13
32056.0
74.4
99.7
70.0
10.0
0132.4
Max
.gro
wth
Thic
knes
sL
ight
Hydro
det
ach
13
4931.7
68.5
43.8
20.1
50.0
96.7
Max
.gro
wth
Thic
knes
sN
utr
ient
Hydro
det
ach
13
4969.5
69.0
54.3
30.1
10.0
78.7
Max
.gro
wth
Thic
knes
sC
ata
det
ach
Hydro
det
ach
13
4916.2
68.3
23.6
00.1
70.1
06.0
Max
.gro
wth
Thic
knes
sC
ata
det
ach
T-C
Hydro
det
ach
13
5916.2
73.8
99.1
70.0
10.0
198.0
Max
.gro
wth
Thic
knes
sC
ata
det
ach
Lig
ht
Hydro
det
ach
13
5885.9
73.4
58.7
30.0
10.0
178.7
Max
.gro
wth
Thic
knes
sC
ata
det
ach
Nutr
ient
Hydro
det
ach
13
5916.2
73.8
99.1
70.0
10.0
198.0
Max
.gro
wth
Thic
knes
sT-C
Lig
ht
Hydro
det
ach
13
5885.9
73.4
58.7
30.0
10.0
178.7
Max
.gro
wth
Thic
knes
sT-C
Nutr
ient
Hydro
det
ach
13
5885.9
73.4
58.7
30.0
10.0
178.7
Max
.gro
wth
Thic
knes
sL
ight
Nutr
ient
Hydro
det
ach
13
5885.9
73.4
58.7
30.0
10.0
178.7
Stu
dy
case
:01-9
Max
.gro
wth
13
11587.5
64.8
32.9
80.2
30.0
74.4
Max
.gro
wth
Thic
knes
s13
21509.0
67.0
05.1
50.0
80.0
213.1
Max
.gro
wth
T-C
13
21587.5
67.6
65.8
10.0
50.0
218.3
Max
.gro
wth
Lig
ht
13
21587.5
67.6
65.8
10.0
50.0
218.3
Max
.gro
wth
Nutr
ient
13
21537.1
67.2
55.3
90.0
70.0
214.8
Max
.gro
wth
Hydro
det
ach
13
21303.6
65.1
03.2
50.2
00.0
65.1
Max.
gro
wth
Cata
det
ach
13
21015.2
61.85
0.00
1.0
00.3
21.0
Max
.gro
wth
Thic
knes
sH
ydro
det
ach
13
31303.6
68.5
76.7
20.0
30.0
128.7
Max
.gro
wth
Thic
knes
sC
ata
det
ach
13
3910.7
63.9
12.0
50.3
60.1
12.8
Max
.gro
wth
Thic
knes
sT8C
13
31509.0
70.4
78.6
20.0
10.0
074.4
Max
.gro
wth
Thic
knes
sL
ight
13
31509.0
70.4
78.6
20.0
10.0
074.4
Max
.gro
wth
Thic
knes
sN
utr
ient
13
31509.0
70.4
78.6
20.0
10.0
074.4
(Conti
nues
)
Tab
leII
I.(C
onti
nued
)
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
A MINIMAL ADEQUATE MODEL FOR RIVER EPILITHON BIOMASS
Tab
leII
I.(C
onti
nued
)
Act
ivat
edpro
cess
esn
pR
SS
AIC
cD
AIC
cex
p(-D
AIC
c/2)
vi
vj/v
i
Max
.gro
wth
T8C
Cat
adet
ach
13
31015.2
65.3
23.4
70.1
80.0
65.7
Max
.gro
wth
Lig
ht
Hydro
det
ach
13
31303.6
68.5
76.7
20.0
30.0
128.7
Max
.gro
wth
Lig
ht
Cat
adet
ach
13
31015.2
65.3
23.4
70.1
80.0
65.7
Max
.gro
wth
Nutr
ient
Hydro
det
ach
13
31303.6
68.5
76.7
20.0
30.0
128.7
Max
.gro
wth
Nutr
ient
Cat
adet
ach
13
31015.2
65.3
23.4
70.1
80.0
65.7
Max
.gro
wth
Cat
adet
ach
Hydro
det
ach
13
31015.2
65.3
23.4
70.1
80.0
65.7
Max
.gro
wth
Thic
knes
sT8C
Cat
adet
ach
13
4910.7
68.2
46.3
90.0
40.0
124.4
Max
.gro
wth
Thic
knes
sL
ight
Cat
adet
ach
13
4910.7
68.2
46.3
90.0
40.0
124.4
Max
.gro
wth
Thic
knes
sN
utr
ient
Cat
adet
ach
13
4910.7
68.2
46.3
90.0
40.0
124.4
Max
.gro
wth
Thic
knes
sC
ata
det
ach
Hydro
det
ach
13
4910.7
68.2
46.3
90.0
40.0
124.4
Max
.gro
wth
T8C
Lig
ht
Cat
adet
ach
13
41015.2
69.6
57.8
00.0
20.0
149.4
Max
.gro
wth
T8C
Nutr
ient
Cat
adet
ach
13
41015.2
69.6
57.8
00.0
20.0
149.4
Max
.gro
wth
T8C
Hydro
det
ach
Cat
adet
ach
13
41015.2
69.6
57.8
00.0
20.0
149.4
Max
.gro
wth
Lig
ht
Nutr
ient
Cat
adet
ach
13
41015.2
69.6
57.8
00.0
20.0
149.4
Max
.gro
wth
Lig
ht
Hydro
det
ach
Cat
adet
ach
13
41015.2
69.6
57.8
00.0
20.0
149.4
Max
.gro
wth
Nutr
ient
Hydro
det
ach
Cat
adet
ach
13
41015.2
69.6
57.8
00.0
20.0
149.4
The
firs
tco
lum
nin
dic
ates
acti
vat
edpro
cess
es(a
nd
thus
acti
vat
edpar
amet
ers)
:m
ax.gro
wth
form
max,0
acti
vat
ion;
Hydro
det
ach
for
c detac
tivat
ion;
Cat
adet
ach
for
k catac
tivat
ion;
Thic
knes
sfo
rk i
nv,B
acti
vat
ion;
Lig
ht
for
k Iac
tivat
ion;
tem
per
ature
forb
acti
vat
ion
and
Nutr
ient
for
k Pac
tivat
ion.
nis
the
num
ber
of
obse
rved
dat
a.p
isth
enum
ber
of
par
amet
erof
each
-sub-m
odel
.R
SS
corr
esponds
toth
ere
sidual
sum
of
squar
esbet
wee
nobse
rved
and
sim
ula
ted
dat
aof
each
sub-m
odel
.AIC
cis
the
Akai
ke
info
rmat
ion
crit
erio
nad
just
edfo
rsm
allsa
mple
size
s.D
AIC
cin
dic
ates
the
amountof
support
for
the
sub-m
odel
rela
tive
toth
eto
p-r
ankin
gone.
Akai
ke
wei
ght(v
i)is
anoth
erin
dex
of
the
stre
ngth
of
evid
ence
for
each
sub-m
odel
.Itis
the
rati
obet
wee
nD
AIC
cof
the
targ
etsu
b-m
odel
rela
tive
toal
lth
eoth
ersu
b-m
odel
san
dca
nbe
inte
rpre
ted
asth
epro
bab
ilit
yof
the
sub-m
odel
bei
ng
corr
ect
giv
enth
edat
a.v
j/v
iis
the
evid
ence
rati
osh
ow
ing
the
exte
nt
tow
hic
hth
e‘b
est’
sub-m
odel
isbet
ter
than
the
model
inques
tion.A
Di<
2(o
ran
evid
ence
rati
o<
2.7
)su
gges
tssu
bst
anti
alev
iden
cefo
rth
esu
b-m
odel
.E
ver
yca
sew
her
eD
i>
10,i
ndic
atin
gth
atth
em
odel
isver
yunli
kel
y,w
asdel
eted
from
this
table
.V
alues
corr
espondin
gto
the
min
imal
AIC
,to
aD
i<
2an
dto
anev
iden
cera
tio<
2.7
are
inbold
.T
he
sele
cted
‘bes
t’su
b-m
odel
isit
alic
ized
.
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
S. BOULETREAU ET AL.
Figure 1. Biomass of epilithon at different sites along the Aguera stream. Lines correspond to values simulated with the best model (minimalAIC), dots to observed biomass (� SE). Models selected for sites 5 and 7, period 2001–02 not shown as too simple (exponential increase) to be
relevant
A MINIMAL ADEQUATE MODEL FOR RIVER EPILITHON BIOMASS
cases, the ‘best’ sub-model (with the smallest AICc value) was the 3-parameter model (mmax,0, kinv,B and cdet). The
‘best’ sub-model in case 01–9 included mmax,0 and kcat. In case 01–2, the Akaike weight and the evidence
ratio indicated that three sub-models (2 and 3 parameters) performed similarly. These included mmax,0, cdet
and also kinv,B. In cases 01–4 and 01–5, the simplest sub-model (mmax,0) had the minimal AICc. Thus, 9 of
11 selected sub-models described epilithon biomass dynamics as the equilibrium between growth and
discharge-dependent loss terms. However, including parameters such as b, kI or kP did not result in significant
improvements on AICc values in any of the cases studied. Nevertheless, RSS decreased by inclusion of b in 90–2;
by inclusion of kinv,B and kcat in 01–4; by inclusion of kinv,B and kcat and kI in 01–5; by inclusion of kI and kcat in 01–7
and by inclusion of kinv,B in 01–9. The fits produced with the best and most convenient sub-model are illustrated in
Figure 1. Simulations combining the net growth term limited by biomass thickness and discharge correctly
reproduced the global pattern described by epilithon dynamics in 1990 and 1992. On the other hand, this model
fitted worse in 2001.
Step-wise regressions were performed on log-transformed data from six situations (90–2, 90–4, 90–5, 90–7,
01–2 and 01–7) corresponding to the best sub-model (mmax,0, kinv,B and cdet). The cases 92–5 and 92–9 were
excluded from the analyses because no physico-chemical data were available in 1992. Results showed that mmax,0
was negatively correlated with the percentage of canopy cover, while the other variables, particularly nutrients,
were excluded from the regression (Table IV), suggesting that epilithon maximal growth rate is controlled by
canopy cover (R2¼ 0.86, p¼ 0.007). No significant correlation was found between kinv,B or cdet and the
environmental variables tested.
DISCUSSION
The dataset presented in Elosegi and Pozo (1998) and Izagirre and Elosegi (2005) was a good candidate to check the
biological realism of the model developed by Uehlinger et al. (1996) under contrasting conditions. The monthly
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
Table IV. Results of step-wise multiple regression analysis between environmental factors (mean velocity, sand %, watertemperature, conductivity, total nitrogen, total phosphorus, silicate and canopy cover %) and parameters of the best sub-models(mmax,0, kinv,B and cdet) of study cases 90–2, 90–4, 90–5, 90–7, 01–2 and 01–7
Dependent variable Independent variable Coefficient Standard error t-value Significativity level
mmax,0 Constant 0.69 0.09 7.54 0.001Canopy cover �0.31 0.062 �5.02 0.007
kinv,B Constant 0.27 0.09 2.89 0.03cdet Constant 0.08 0.02 3.44 0.02
S. BOULETREAU ET AL.
sampling frequency is compatible with the application of models with few parameters allowing only major trends to
be observed and thus emphasizing the role of major environmental factors. The dataset included three 12-month
data series of epilithon biomass sampled at five sites along the channel of the Aguera stream, selected for the
broadest range of environmental conditions. As flood frequency and intensity are known to radically change from
year to year, contrasting hydrological contexts were observed. The sampling year 1990 was characterized by low
discharge (lower than 30 m3 s�1) and a 6-month flood-free period lasting from May to October, which allowed for
an important development of epilithon biomass that peaked at 300 g AFDM m�2. In sampling periods 1992–93 and
2001–02 more frequent floods and higher discharge were recorded, especially in 1992, and peak biomass reached
only 50 g AFDM m�2. Analysis of this dataset concluded hydrology to be the major controlling factor, except at
canopy covered sites where light availability overrides the effect of floods (Izagirre and Elosegi, 2005).
Ecological modelling strives to identify models that capture the essence of a system, explaining observations and
perhaps ultimately permitting prediction. Simple models contain fewer nuisance variables and have greater
generality (Ginzburg and Jensen, 2004). For that reason our aim was to identify a simple model that was in general
agreement with observed data. We used model fitting to investigate the most appropriate and simplest model form
to describe epilithon dynamics. Models with simplicity, parsimony and minimum adequacy based on Occam’s
Razor have recently been promoted in theoretical and applied ecology (Burnham and Anderson, 2001; Johnson and
Omland, 2004). With this objective, model selection was performed by applying a global optimization algorithm to
fit mechanistic non-linear models to time-series data, and AIC to check agreement between modelled and observed
data. AIC and related criteria (e.g. the Bayesian information criterion) provide an alternative to traditional analyses
for evaluating variable or parameter combinations, especially in studies with few hypotheses and a small sample
size. AIC and its related measures were first applied almost exclusively in the context of model selection in
capture-recapture analyses (Lebreton et al., 1992; Anderson et al., 1994) but in the past decade have been
recognized as a valuable tool in more general situations (Johnson and Omland, 2004). AIC is a likelihood-based
measure of model fit that accounts for the number of parameters estimated in a model (Akaike, 1973). It has two
components: negative log-likelihood, which measures lack of model fit to the observed data, and a bias correction
error, which increases as a function of the number of model parameters. Models with a large number of parameters
are penalized more heavily than those with a small number of parameters. Therefore, the model with the lowest AIC
has the best relative fit. If only poor models are considered, the AIC will select the best of the poor models,
highlighting the importance of determining the set of suitable candidate models with respect to the current
knowledge of the modelled processes.
According to AIC results, 7 simulations of 11 in the present study showed that modelling proved to give a correct
fit by means of three parameters (mmax,0, kinv,B and cdet). Epilithon biomass dynamics in the Aguera stream (this
dataset) can be simply simulated by accounting for the same processes as in the river Necker (Uehlinger et al.
dataset). Epilithon biomass dynamics can roughly be considered to be the result of equilibrium between
phototrophic growth and discharge-dependent loss.
The implementation of the density-limited growth parameter (kinv,B) into biomass growth appeared to be
significant in describing epilithon biomass accrual. Light (corrected by accounting for the canopy cover at each
site), temperature and phosphorus limitation processes did not significantly improve the goodness-of-fit and were
much less important than the above-mentioned processes. Although variations in both nutrient conditions and solar
radiation (due to differences in canopy cover) have been used to explain spatial biomass dynamics in the Aguera
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
A MINIMAL ADEQUATE MODEL FOR RIVER EPILITHON BIOMASS
stream (Izagirre and Elosegi, 2005), their contribution to epilithon development was not underlined by our models:
whatever the conditions (site and year), light availability was not a predominant factor explaining temporal biomass
dynamics. In streams with unpredictable flow patterns, light has been reported not to be a significant predictor of
biomass (Fisher and Grimm, 1988; Biggs and Close, 1989). Nevertheless, results of step-wise regression in the
present investigation suggested a linear dependence of maximum biomass growth rate on canopy cover, in
accordance with conclusions from Izagirre and Elosegi (2005). These paradoxical results raise the question of
representation of the growth light limitation term in the equation. Light was assumed to be an ‘interactive-essential’
resource with the limiting nutrient, following Tilman (1982) and Huisman et al. (1999). It means that both nutrient
availability and light limitation affect the primary producers to some extent. Despite its simplicity, the light
limitation term integrating daily irradiance changes was ultimately not adapted to represent spatial light variability
from one site to another. Canopy cover can be considered to be a more adapted variable than the global daily
radiation to represent light limitation of epilithon in the Aguera stream.
Our study confirmed that the dynamics of epilithon biomass in the Aguera stream are mainly driven by
hydrodynamics, as previously stated in reports on streams and rivers (Biggs and Close, 1989; Biggs and Thomsen,
1995; Bouletreau et al., 2006). Biomass detachment mainly requires a sufficient critical shear stress (linked to
discharge) to dislodge material from the epilithon mat (Biggs and Close, 1989). Regarding the value of the
detachment parameter, we did not observe a correlation between cdet (a descriptor of substrate sensitivity to floods)
and the percentage of sand on the river bottom. Assemblages colonizing fine substrata would probably be scoured
more easily by flooding (Cattaneo et al., 1997). Moreover, under low current conditions biofilms tend to grow as
loose assemblages, which are easily scoured by changes in river flow regime (Peterson, 1996). Hence, this result
suggests that cdet cannot simply be seen as an estimate of the biofilm resistance to shear stress. In 1990, several
floods strong enough to slough epilithon off the substrate were recorded and inter-flood periods were sufficiently
long and stable to allow for large biomass accumulation, mainly due to filamentous algae. In 2001–02, the simple
model including biomass-dependent growth rate detachment losses directly proportional to discharge and biomass
was the minimal adequate model, but in some cases its fit was not particularly good. Poor fit between observed and
modelled data can arise from both process and observational errors. Regarding observational errors, the time-series
was small (13 points) and showed rather large measurement errors with coefficients of variation up to 50%,
reflecting a very patchy distribution of epilithon. Regarding process errors, the temporal pattern described by
epilithon in 2001 was much less conspicuous than the clear cycles of biomass accumulation and loss in 1990.
Unpredictable hydrodynamic changes were not the only substantial factor in biomass loss in 2001, suggesting that
the model could be partly insufficient for explaining parameters and predicting biomass dynamics. For example, the
long low-water period that the Garonne River experienced in 2001 required inclusion of an additional autogenic
biomass loss factor in the model (Bouletreau et al., 2006). A similar 6-month flood-free period was observed in the
Aguera stream in 1990, but epilithon biofilm development persisted, attaining very high biomasses without
autogenic sloughing. The latter finding could be attributed to the mat structure being dominated by cyanobacteria,
as many species of these bacteria can grow heterotrophically, fix atmospheric nitrogen and engage in anoxic
photosynthesis. It can thus be hypothesized that cyanobacteria-dominated epilithon displays a higher resistance to
scouring, forming cohesive mats as observed in the Llobregat River (Sabater et al., 2003).
Temporal dynamics of epilithon can be quite contrasting. Biomass can be well described by alternating accrual
(one simple growth function) and one flow-related loss, especially when large floods combine with short and stable
inter-flood periods. If the inter-flood period is sufficiently stable and long, one or several factors such as seasonal
changes in communities, autogenic sloughing, grazing or internal nutrient loading can interact and produce
complex trajectories of epilithic biomass. On the other hand, when floods are too frequent, epilithic patchiness can
mask temporal differences and no clear pattern can be discerned. Nevertheless our results suggest that the model
formulation developed by Uehlinger et al. (1996) would be a satisfactory minimal adequate model to describe the
biomass dynamics of river epilithon in contrasting conditions.
To improve model performance one should increase model transferability, i.e. its ability to be applied in different
conditions at the relevant spatial and temporal scales without requiring changes in model structure or
parameterizations (Snowling and Kramer, 2001). In the present study the model was indeed transferred both
spatially and temporally, as we used the model formulation developed by Uehlinger et al. (1996) in the river Necker
to estimate the epilithic biomass dynamics in the Aguera stream. The model thus transferred worked pretty well,
Copyright # 2007 John Wiley & Sons, Ltd. River. Res. Applic. (2007)
DOI: 10.1002/rra
S. BOULETREAU ET AL.
especially in hydrological conditions that combine contrasting short low with high water periods, but a
re-calibration of parameters was unavoidable to retain accuracy. Transferability is known to be challenging in
ecological models (Leftwich et al., 1997; Guay et al., 2003). Transferability could perhaps improve making the
model more complex, taking into account additional processes to increase flexibility and more sensitivity.
Designing a model that would be transferable is likely to be an open question for modelling the biomass dynamics
of river epilithon. Nevertheless, this would require an adapted dataset with high number and frequency of observed
data.
ACKNOWLEDGEMENTS
Stephanie Bouletreau was supported by FEDER (Fonds Europeens de Developpement Regional) and a grant for
foreign exchange (ATUPS) from the University Paul Sabatier. Oihana Izagirre did part of this work thanks to a
pre-doctoral grant by the Basque Government. This work was supported by the research projects PIGV 8924
(Basque Government), 9/UPV00118.310-14476/2002 (University of the Basque Country), DGICYT PB459/92,
MCYT BOS2003-04466 (Spanish Government) and by the GIS-ECOBAG (Groupement d’Interet Scientifi-
que-Ecologie et Economie du Bassin Adour Garonne). Discharge data were kindly provided by the Spanish
Northern Hydrographical Confederation and solar radiation data by the Spanish Meteorological Institute.
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